Answer is: option3

Solution:

The question is based on the Mean Value Theorem (MVT), which states that if a function \( f(x) \) is:

1. Continuous on the closed interval \([a, b]\), and
2. Differentiable on the open interval \((a, b)\),

then there exists at least one point \( c \in (a, b) \) such that:

\[ f'(c) = \frac{f(b) - f(a)}{b - a} \]

The secant line connects the points \( A(a, f(a)) \) and \( B(b, f(b)) \).

The slope of this secant line is given by:

\[ m = \frac{f(b) - f(a)}{b - a} \]

The theorem guarantees that there will be at least one point where the tangent to \( f(x) \) has the same slope as the secant line.

By inspecting the graph, we need to count how many times the derivative \( f'(x) \) (the slope of the tangent line) equals \( m \).

The graph has multiple peaks and valleys, meaning it crosses the secant slope multiple times.

Conclusion:

Observing the graph, the function appears to be differentiable and crosses the slope of the secant line four times.

Final Answer: 4.